

A068140


Smaller of two consecutive numbers each divisible by a cube greater than one.


19



80, 135, 296, 343, 351, 375, 512, 567, 624, 728, 783, 944, 999, 1160, 1215, 1375, 1376, 1431, 1592, 1624, 1647, 1808, 1863, 2024, 2079, 2240, 2295, 2375, 2400, 2456, 2511, 2624, 2672, 2727, 2888, 2943, 3087, 3104, 3159, 3320, 3375, 3429, 3536, 3591
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OFFSET

1,1


COMMENTS

Cubeful numbers with cubeful successors. This is to cubes as A068781 is to squares. 1375 is the smallest of three consecutive numbers divisible by a cube, since 1375 = 5^3 * 11 and 1376 = 2^5 * 43 and 1377 = 3^4 * 17. What is the smallest of four consecutive numbers divisible by a cube? Of n consecutive numbers divisible by a cube?  Jonathan Vos Post, Sep 18 2007
22624 is the smallest of four consecutive numbers each divisible by a cube, with factorizations 2^5 * 7 * 101, 5^3 * 181, 2 * 3^3 * 419, and 11^3 * 17.  D. S. McNeil, Dec 10 2010
18035622 is the smallest of five consecutive numbers each divisible by a cube. 4379776620 is the smallest of six consecutive numbers each divisible by a cube. 1204244328624 is the smallest of seven consecutive numbers each divisible by a cube.  Donovan Johnson, Dec 13 2010
The sequence is the union, over all pairs of distinct primes (p,q), of numbers == 0 mod p^3 and == 1 mod q^3 or vice versa.  Robert Israel, Aug 13 2018
The asymptotic density of this sequence is 1  2/zeta(3) + Product_{p prime} (1  2/p^3) = 1  2 * A088453 + A340153 = 0.013077991848467056243...  Amiram Eldar, Feb 16 2021


LINKS



FORMULA



EXAMPLE

343 is a term as 343 = 7^3 and 344= 2^3 * 43.


MAPLE

isA068140 := proc(n)
isA046099(n) and isA046099(n+1) ;
end proc:
for n from 1 to 4000 do
if isA068140(n) then
printf("%d, ", n) ;
end if;


MATHEMATICA

a = b = 0; Do[b = Max[ Transpose[ FactorInteger[n]] [[2]]]; If[a > 2 && b > 2, Print[n  1]]; a = b, {n, 2, 5000}]
Select[Range[2, 6000], Max[Transpose[FactorInteger[ # ]][[2]]] > 2 && Max[Transpose[FactorInteger[ # + 1]][[2]]] > 2 &] (* Jonathan Vos Post, Sep 18 2007 *)
SequencePosition[Table[If[AnyTrue[Rest[Divisors[n]], IntegerQ[Surd[#, 3]]&], 1, 0], {n, 3600}], {1, 1}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 18 2020 *)


CROSSREFS

Cf. A046099, A063528, A068781, A068782, A068783, A068784, A088453, A122692, A174113, A340152, A340153.


KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



